scales of motion, reynolds averaging september 22

Scales of Motion, Scales of Motion, Reynolds Reynolds averagingaveraging

September 22September 22

xFvxP

zuw

yuv

xuu

tu

sin21

Recall the momentum Recall the momentum equation for a fluid:equation for a fluid:

Fx is a frictional forcer per unit mass

This includes all friction, including winds

Figure 8.1 in Stewart

Frictional StressesFrictional Stresses Internal stress (frictional stress)Internal stress (frictional stress) Molecular friction stress related to Molecular friction stress related to

velocity shearvelocity shear

zuTxz

Molecular Viscosity

uF

zu

yu

xuF

x

x

2)(

2

2

2

2

2

2)( ;

del squareKinematic molecular

viscosity

uy

x

02

2

yu

constant stress, no

acceleration in fluid

uy

x

02

2

yu

changing stress,

acceleration of fluid

“curvature of velocity field”

zT

yT

xT

zu

zyu

yxu

xF xzxyxxx

1

For an incompressible fluid, the frictional force per unit mass takes the form:

TurbulenceTurbulence Turbulence comes from the non-linear terms Turbulence comes from the non-linear terms

in the momentum equationsin the momentum equations

ufvxp

zuw

yuv

xuu

tu

dtdu 21

local advective, non-linear

Reynolds NumberReynolds Number Reynolds Number Reynolds Number RRee, is the ratio of the , is the ratio of the

non-linear terms to the viscous termsnon-linear terms to the viscous terms

UL

LULUU

xuxuu

ViscouslinearNonRe

22

2

U: typical velocity scale u and ∂u ≈ U

L: typical distance scale ∂x ≈ L

LU

xuu

2

22

2

LU

xu

UL

Pipe flow

U = average velocity in center

L = radius of pipe-Flow is not turbulent (non-linear terms are not important) if Re<1000 in pipes

Turbulence Turbulence In more complicated flows, non-linear terms are In more complicated flows, non-linear terms are

important when Rimportant when Ree≈ 1 or larger≈ 1 or larger Flows become fully turbulent if RFlows become fully turbulent if Ree>10>1055

Gulf Stream: U~1 m/s L~100 kmGulf Stream: U~1 m/s L~100 km ν≈ν≈1010-6 -6 mm22ss-1 -1 RRee≈10≈101111 => turbulent => turbulent flowflow

Non-linear effects are strong compared to Non-linear effects are strong compared to molecular frictionmolecular friction

In open ocean, molecular friction is usually In open ocean, molecular friction is usually ignored! Turbulent (non-linear) terms are the ignored! Turbulent (non-linear) terms are the important ones!important ones!

Turbulent fluctuations act like molecular friction – Turbulent fluctuations act like molecular friction – dissipate (redistribute) energy and other propertiesdissipate (redistribute) energy and other properties

Turbulent stressesTurbulent stresses

pppwwwvvvuuu

mean over some period

“eddy”

0uby definition

T

T

TuTudt

tu

Ttu

dttuT

uu

0

0

)0()(1

)(1

Non-linear terms in the momentum equation can be written as:

xuu

xuu

xuuuu

xuu

xuu

xuu

xuu

xuuuu

Using the previous equation, the continuity equation splits into two equations

0

0

xw

xv

xu

dxw

dxv

dxu

The x-component of the of the momentum equation becomes

wuzu

zvu

yu

yuu

xu

x

vxP

DtuD

sin21

At large Reynolds number, is very small compared to other terms. The addition force per unit mass due to turbulence is:

wuz

vuy

uux

Fx

Reynolds Stresses

Reynolds StressesReynolds Stresses

we assume turbulence acts like molecular

viscocity:

zuATwu zxz

kinematic eddy viscosity

2

2

zuA

zuA

zzT

zzxz

assume that Az is either constant or that it varies more slowly in the z direction than ∂ū/∂z. Similar for horizontal components but Az is much less than Ax,y=AH


The buoyancy force acting on the displaced parcel is:

2gVF ‘

The acceleration of the displaced parcel is:

)( 2g

mFa

Stability EquationStability Equation

dzdE

1

Stability is defined such that:

E > 0 stable

E = 0 neutral stability

E < 0 unstable

Influence of stability is expressed by a stability frequency N

(also known as Brunt-Vaisala frequency):gEN 2

Richardson NumberRichardson Number The relative importance of static stability The relative importance of static stability

and dynamic instability is expressed by and dynamic instability is expressed by the Richardson Number:the Richardson Number:

2zugERi

Ri > 0.25 Stable

Ri < 0.25 Velocity shear enhances turbulence

scales of motion, reynolds averaging september 22

Documents